How to mirror the contraints rotation and limit rotation?
To do what you want it will be easier to use python. I wrote a script that copies constraint from bones with ".L" to corresponding bones with ".R"

import bpy
context = bpy.context
scene = context.scene
arm = bpy.data.objects["Armature"]
bpy.ops.object.select_all(action='DESELECT')
arm.select = True
bpy.context.scene.objects.active = arm
bpy.ops.object.posemode_toggle()
bpy.ops.pose.select_all(action='SELECT')
all_bones = bpy.context.selected_pose_bones
for bone in all_bones:
    bpy.ops.pose.select_all(action='DESELECT')
    if ".L" in bone.name:
        boneL = arm.data.bones[bone.name]
        boneR = arm.data.bones[bone.name.replace('.L','.R')]
        boneR.select = True
        boneL.select = True
        arm.data.bones.active = boneL
        bpy.ops.object.constraints_copy()
        bpy.ops.pose.constraints_copy()
        for con in arm.pose.bones[boneR.name].constraints:
           con.subtarget = con.subtarget.replace('.L','.R')


Another way to do it without Python is to use the Copy Constraint, which involves manual work to be done, you select an object then add a Copy Constraint, then select which other object you want to copy the property from (location, rotation). So no need for paste constraint.